While geostrophic motion refers to the wind that would result from an exact balance between the Coriolis force and horizontal pressure-gradient forces,[1] quasi-geostrophic (QG) motion refers to flows where the Coriolis force and pressure gradient forces are almost in balance, but with inertia also having an effect.
[2] Atmospheric and oceanographic flows take place over horizontal length scales which are very large compared to their vertical length scale, and so they can be described using the shallow water equations.
The Rossby number is a dimensionless number which characterises the strength of inertia compared to the strength of the Coriolis force.
The quasi-geostrophic equations are approximations to the shallow water equations in the limit of small Rossby number, so that inertial forces are an order of magnitude smaller than the Coriolis and pressure forces.
If the Rossby number is equal to zero then we recover geostrophic flow.
The quasi-geostrophic equations were first formulated by Jule Charney.
[3] In Cartesian coordinates, the components of the geostrophic wind are where
The geostrophic vorticity can therefore be expressed in terms of the geopotential as Equation (2) can be used to find
The quasi-geostrophic vorticity equation can be obtained from the
components of the quasi-geostrophic momentum equation which can then be derived from the horizontal momentum equation
The material derivative in (3) is defined by The horizontal velocity
Two important assumptions of the quasi-geostrophic approximation are
The second assumption justifies letting the Coriolis parameter have a constant value
[4] However, because the acceleration following the motion, which is given in (1) as the difference between the Coriolis force and the pressure gradient force, depends on the departure of the actual wind from the geostrophic wind, it is not permissible to simply replace the velocity by its geostrophic velocity in the Coriolis term.
The approximate horizontal momentum equation thus has the form
Expressing equation (7) in terms of its components,
, and noting that geostrophic wind is nondivergent (i.e.,
) and that the divergence of the ageostrophic wind can be written in terms of
equation (9) can therefore be written as Defining the geopotential tendency
and noting that partial differentiation may be reversed, equation (10) can be rewritten in terms of
The right-hand side of equation (11) depends on variables
Using the chain rule of differentiation, term C can be written as
But based on the thermal wind relation,
can be expressed in the form of a conservation equation [6]
is the quasi-geostrophic potential vorticity defined by
As an air parcel moves about in the atmosphere, its relative, planetary and stretching vorticities may change but equation (17) shows that the sum of the three must be conserved following the geostrophic motion.
Alternatively, it can also be used to predict the evolution of the geopotential field given an initial distribution of
and suitable boundary conditions by using an inversion process.
More importantly, the quasi-geostrophic system reduces the five-variable primitive equations to a one-equation system where all variables such as
, the vorticity equation can be used to diagnose vertical motion provided that the fields of both